ffsim.contract

Functions for contracting tensors and constructing linear operators.

ffsim.contract.contract_diag_coulomb(vec, mat, norb, nelec, *, z_representation=False)

Contract a diagonal Coulomb operator with a vector.

A diagonal Coulomb operator has the form

\[\sum_{i, j, \sigma, \tau} Z^{(\sigma \tau)}_{ij} n_{\sigma, i} n_{\tau, j} / 2\]

where \(n_{\sigma, i}\) denotes the number operator on orbital \(i\) with spin \(\sigma\) and \(Z^{(\sigma \tau)}\) is a real-valued matrix

Parameters:

• vec (ndarray) – The state vector to be transformed.

• mat (ndarray | tuple[ndarray | None, ndarray | None, ndarray | None]) – The diagonal Coulomb matrix \(Z\). You can pass either a single Numpy array specifying the coefficients to use for all spin interactions, or you can pass a tuple of three Numpy arrays specifying independent coefficients for alpha-alpha, alpha-beta, and beta-beta interactions (in that order). If passing a tuple, you can set a tuple element to None to indicate the absence of interactions of that type.

• norb (int) – The number of spatial orbitals.

• nelec (tuple[int, int]) – The number of alpha and beta electrons.

• z_representation (bool) – Whether the input matrices are in the “Z” representation.

Return type:

ndarray

Returns:

The result of applying the diagonal Coulomb operator on the input state vector.

ffsim.contract.contract_num_op_sum(vec, coeffs, norb, nelec)

Contract a linear combination of number operators with a vector.

A linear combination of number operators has the form

\[\sum_{\sigma, i} \lambda_i n_{\sigma, i}\]

where \(n_{\sigma, i}\) denotes the number operator on orbital \(i\) with spin \(\sigma\) and the \(\lambda_i\) are real numbers.

Parameters:

• vec (ndarray) – The state vector to be transformed.

• coeffs (ndarray) – The coefficients of the linear combination.

• norb (int) – The number of spatial orbitals.

• nelec (tuple[int, int]) – The number of alpha and beta electrons.

Returns:

The result of applying the linear combination of number operators on the input state vector.

ffsim.contract.contract_one_body(vec, mat, norb, nelec)

Contract a one-body tensor with a vector.

A one-body tensor has the form

\[\sum_{ij} M_{ij} a^\dagger_i a_j\]

where \(M\) is a complex-valued matrix.

Parameters:

• mat (ndarray) – The one-body tensor.

• norb (int) – The number of spatial orbitals.

• nelec (tuple[int, int]) – The number of alpha and beta electrons.

Return type:

ndarray

Returns:

A LinearOperator that implements the action of the one-body tensor.

ffsim.contract.diag_coulomb_linop(mat, norb, nelec, *, orbital_rotation=None, z_representation=False)

Convert a (rotated) diagonal Coulomb matrix to a linear operator.

A rotated diagonal Coulomb operator has the form

\[\mathcal{U} (\sum_{i, j, \sigma, \tau} Z^{(\sigma \tau)}_{ij} n_{\sigma, i} n_{\tau, j} / 2) \mathcal{U}^\dagger\]

where \(n_{\sigma, i}\) denotes the number operator on orbital \(i\) with spin \(\sigma\), \(Z^{(\sigma \tau)}\) is a real-valued matrix, and \(\mathcal{U}\) is an optional orbital rotation.

Parameters:

• mat (ndarray | tuple[ndarray | None, ndarray | None, ndarray | None]) – The diagonal Coulomb matrix \(Z\). You can pass either a single Numpy array specifying the coefficients to use for all spin interactions, or you can pass a tuple of three Numpy arrays specifying independent coefficients for alpha-alpha, alpha-beta, and beta-beta interactions (in that order). If passing a tuple, you can set a tuple element to None to indicate the absence of interactions of that type.

• norb (int) – The number of spatial orbitals.

• nelec (tuple[int, int]) – The number of alpha and beta electrons.

• orbital_rotation (ndarray | tuple[ndarray | None, ndarray | None] | None) – The optional orbital rotation. You can pass either a single Numpy array specifying the orbital rotation to apply to both spin sectors, or you can pass a pair of Numpy arrays specifying independent orbital rotations for spin alpha and spin beta. If passing a pair, you can use None for one of the values in the pair to indicate that no operation should be applied to that spin sector.

• mat_alpha_beta – A matrix of coefficients to use for interactions between orbitals with differing spin.

• z_representation (bool) – Whether the input matrices are in the “Z” representation.

Return type:

LinearOperator

Returns:

A LinearOperator that implements the action of the diagonal Coulomb operator.

ffsim.contract.num_op_sum_linop(coeffs, norb, nelec, *, orbital_rotation=None)

Convert a (rotated) linear combination of number operators to a linear operator.

A rotated linear combination of number operators has the form

\[\mathcal{U} (\sum_{\sigma, i} \lambda_i n_{\sigma, i}) \mathcal{U}^\dagger\]

where \(n_{\sigma, i}\) denotes the number operator on orbital \(i\) with spin \(\sigma\), the \(\lambda_i\) are real numbers, and \(\mathcal{U}\) is an optional orbital rotation.

Parameters:

• coeffs (ndarray) – The coefficients of the linear combination.

• norb (int) – The number of spatial orbitals.

• nelec (tuple[int, int]) – The number of alpha and beta electrons.

• orbital_rotation (ndarray | None) – A unitary matrix describing the optional orbital rotation.

Return type:

LinearOperator

Returns:

A LinearOperator that implements the action of the linear combination of number operators.

ffsim.contract.one_body_linop(mat, norb, nelec)

Convert a one-body tensor to a linear operator.

A one-body tensor has the form

\[\sum_{ij} M_{ij} a^\dagger_i a_j\]

where \(M\) is a complex-valued matrix.

Parameters:

• mat (ndarray) – The one-body tensor.

• norb (int) – The number of spatial orbitals.

• nelec (tuple[int, int]) – The number of alpha and beta electrons.

Return type:

LinearOperator

Returns:

A LinearOperator that implements the action of the one-body tensor.